There are a couple of options I see after performing a quick forums search.
Are you by chance outside the US, if so, try setting the Launcher to use the nearest proxy server.
Check to make sure that gameclient.exe is set to run as Administrator.
Check C:\Users\Public\Games\Cryptic Studios for a text file called errors. Do you see entries such as:
100724 22:42:02 32  ESC : The patchclientlib is internally recording error code : 36\nA file was locked that could not be open for writing. - Unable to read back file C:/Users/Public/Games/Cryptic Studios/Star Trek Online/Live/Microsoft.VC80.CRT/Microsoft.VC80.CRT.manifest!
If so, try disabling your A/V autoprotect, or possibly try deleting those files. If you are using Panda Anti-Virus, it is known to cause game file corruption.
Are you running any games that install nVidia PhysX crap? These craptacular nVidia infestations comes with any game with the nVidia, the way it's meant to be played crap on it. These install when you install the game but can cause issues with ATI hardware in some games, STO included. If you have any nVidia PhysX crap on your system, remove it and try running the game again. Rolling back to the 11.3's also is a good idea as these are the best available drivers for STO on ATI based hardware.
The problem has gotten worse since I messed with my drivers. The game was crashing even at the lowest graphics settings relatively often when I rolled back to 11.3, so I re-updated to 11.5 and had no improvement from the 11.3 drivers.
It's still somewhat playable, but I will often crash once or twice in a ground mission now. I still have not had a space crash since the one time I mentioned earlier in this thread.